A K3 surface is a complex surface 
whose underlying smooth manifold
is a compact manifold, whose canonical
bundle is trivial, and for which 
. Equivalently, it is a simply
connected complex surface that is a compact
manifold and has a nowhere-vanishing holomorphic
differential form that is a 2-form. A complex K3 surface
is a two-dimensional Calabi-Yau space and also
a hyper-Kähler manifold.
The Betti numbers and Euler characteristic of every complex K3 surface are
 |
Its Hodge numbers satisfy 
and 
.
Every smooth quartic surface in complex projective space 
is a K3 surface. Other standard examples include double
covers of the projective plane whose branch
locus is a smooth sextic and Kummer surfaces,
obtained by resolving the 16 singularities of the
quotient space of a two-dimensional complex
torus by the involution 
.
